By Zbigniew H. Nitecki

*Calculus Deconstructed* is an intensive and mathematically rigorous exposition of single-variable calculus for readers with a few past publicity to calculus strategies yet to not equipment of evidence. This e-book is suitable for a starting Honors Calculus direction assuming highschool calculus or a "bridge path" utilizing basica research to encourage and illustrate mathematical rigor. it might probably function a mix textbook and reference publication for person self-study. normal themes and methods in single-variable calculus are offered in context of a coherent logical constitution, construction on typical houses of actual numbers and educating equipment of evidence by way of instance alongside the best way. various examples make stronger either useful and theoretical knowing, and broad historic notes discover the arguments of the originators of the subject.

No earlier adventure with mathematical facts is believed: rhetorical concepts and methods of facts (*reductio advert absurdum*, induction, contrapositives, etc.) are brought through instance alongside the best way. among the textual content and workouts, proofs can be found for the entire simple result of calculus for features of 1 genuine variable.

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**Example text**

E) Use the previous two parts of this exercise to show that if {xk } is bounded and x is its one and only accumulation point, then xk → x. 46 CHAPTER 2. SEQUENCES AND THEIR LIMITS n 31. Let {xk }∞ k=1 be a sequence of real numbers, and let Sn := k=1 xk be ∞ the sequence of partial sums for the series k=1 xk . 10). , there is some α > 0 such that xk = α for all k) then Sn → ∞. (b) Show that if the sequence is (not necessarily constant, but) bounded below by some strictly positive number (xk ≥ α > 0 for some α), then Sn → ∞.

We have just established that if 2 is rational, then there is an √ integer q which is odd and even at the same time. That’s absurd! 3 √ OK, that’s settled. 2 is definitely not rational. How, then, do we locate it on the number line? The best we can do is to devise a scheme that lets us √ position 2 relative to the rationals (whose location we can get exactly). We shall consider two schemes to do this. √ The first—and most familiar one—is to find a decimal expansion for 2. ) we conclude from Exercise 11√that 2 lies between 1 and 2.

A finite sequence of n real numbers is essentially an n-tuple, but to consider limits we need infinite sequences, so whenever we say “sequence”, we mean that it is infinite unless it is explicitly called finite. Some sequences are understood once we establish a pattern with the first few terms. For example, 1, 2, 3, 4, . . is normally recognized as the sequence of positive integers, while 1 1 1 1 1 , , , , ,... 1 2 4 8 16 is understood to be the reciprocals of powers of 2. This is, however, an unreliable method; we prefer a specification that explicitly tells us how to find the value of each number in the sequence.

### Calculus deconstructed: a second course in first-year calculus by Zbigniew H. Nitecki

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